Pis'ma v ZhETF, vol. 97, iss. 7, pp. 496-505
© 2013 April 10
ПО ИТОГАМ ПРОЕКТОВ РОССИЙСКОГО ФОНДА ФУНДАМЕНТАЛЬНЫХ ИССЛЕДОВАНИЙ Проект РФФИ # 10-02-00193
Magneto-optics of monolayer and bilayer graphene
L. A. Falkovsky
Landau Institute for Theoretical Physics of the RAS, 119334 Moscow, Russia Verechagin Institute of the High Pressure Physics of the RAS, 142190 Troitsk, Russia
Submitted 11 March 2013
The optical conductivity of graphene and bilayer graphene in quantizing magnetic fields is studied. Both dynamical conductivities, longitudinal and Hall's, are analytically evaluated. The conductivity peaks are explained in terms of electron transitions. Correspondences between the transition frequencies and the magneto-optical features are established using the theoretical results. The main optical transitions obey the selection rule with An = 1 for the Landau number n. The Faraday rotation and light transmission in the quantizing magnetic fields are calculated. The effects of temperatures and magnetic fields on the chemical potential are considered.
DOI: 10.7868/S0370274X13070114
I. Introduction. The most accurate investigation of the band structure of metals and semiconductors is a study of the Landau levels through experiments such as magneto-optics [1-9] and magneto-transport [10-14]. In magnetic fields, the classical and quantum Hall effects are observed, as well as the polarization rotation for transmitted (Faraday's rotation) or reflected (Kerr's rotation) lights. However, the interpretation of the experimental results involves a significant degree of uncertainty, because it is not clear how the resonances can be identified and which electron transitions they correspond to.
Comprehensive literature on the graphene family can be described in terms of the Dirac gapless fermions. According to this picture, in graphene, there are two bands at the K hexagon vertexes of the Brillouin zone without any gap between them, and the electron dispersion can be considered as linear in the wide wave-vector region. For the dispersion linearity, this region should be small compared with the size of the Brillouin zone, i.e. less than 108 cm-1, providing the small carrier concentration n0 ^ 1016 cm-2. Pristine graphene at zero temperature has no carriers, and the Fermi level should divide the conduction and valence bands. However, undoped graphene cannot be really obtained, and so far purest graphene contains about n0 ~ 109 cm-2 of carriers. Then the following problem appears - how do
Coulomb electron-electron interactions renormalize the linear dispersion and does graphene become an insulator with a gap?
Semiconductors with the gap are needed for electronic applications. Investigations of the graphene bilayer and multilayer are very popular as the gap appears when the bias is applied. Here Slonczewski, Weiss, and McClure (SWMC) should be mentioned because they have stated the description of a layered matter [15] with interactions strong in a layer and weak between layers. The theory contains several parameters which are the hopping integrals for nearest neighbors. Such a picture has been examined in many experiments [16].
The theoretical solution for the band problem in magnetic fields often cannot be exactly found. A typical example is presented by graphene layers. For bilayer graphene and graphite, the effective Hamiltonian is a 4 x 4 matrix giving four energy bands. The trigonal warping described by the small parameter 73 in the effective Hamiltonian provides an evident effect. Another important parameter is the gate-tunable bandgap U in bilayer graphene. In this situation, the quantization problem cannot be solved within a rigorous method. To overcome this difficulty, several methods have been proposed for approximate [8,17-21], numerical [22-25], and semiclassical quantization [26-29].
In this paper, our attention is focused on the dynamic conductivity of monolayer and bilayer graphene in the presence of a constant magnetic field in z-direction. We consider the collisionless limit when the electron collision rate is much less then the frequency of the electric field. Then, the accurate theoretical results can be obtained for Faraday's rotation and trans-mittance through graphene layers. The present paper is organized as follows. In Sec. II we recall the electron dispersion in the monolayer and bilayer graphene. In Sec. III we describe in detail the quantization in magnetic fields. In Sec. IV the longitudinal and Hall conductivities as well as the Faraday rotation are described. Effects of temperatures and magnetic fields on the chemical potential are considered in Sec. V. Section VI contains a summary of the discussed results.
II. Electron dispersion in monolayer and bilayer graphene. Electron dispersion in graphene. The symmetry of the K point is C3v with the threefold axis and reflection planes. This group has twofold representation with the basis functions transforming each in other under reflections and obtaining the factors exp (±2ni/3) in rotations. The linear momentum displacements from the K point, taken as p± = T'i-Px — py, transform in a similar way. The effective Hamiltonian is invariant under the group transformations, and we have the unique possibility to construct the invariant Hamiltonian linear in the momentum as
H (p)
0 vp+ vp— 0
(1)
where v is a constant of the velocity units. The same Hamiltonian can be written using the tight-binding model.
The eigenvalues of this matrix give two bands
£1,2 = Tvypl + P2V = TvP,
where the subscript s = 1,2 numerates these two bands (holes and electrons). The gapless linear spectrum arises as a consequence of the symmetry, and the Fermi energy coincides with the band crossing (the Dirac point) due to the carbon valence. The cyclotron mass has the form
m(e)
1 dS (e) e
2n de
and the carrier concentration at zero temperature n(eF) = eF/nh2v2 is simply expressed in terms of the Fermi energy £F.
Tuning the gate voltage, the linearity of the spectrum has been examined in the Schubnikov-de Haas studies [30] with the help of the connection between
the effective mass and the carrier concentration at the Fermi level m(ep)v = ^h\Jim{£-p). The "constant" parameter v was found to be no longer constant. At low carrier concentrations n ~ 109 cm~2, it exceeds its constant value v = (1.05 ± 0.1) • 108 cm/s for concentrations n > 1011 cm~2 by the factor of 3.
This is a result of electron-electron interactions which become stronger at low carrier concentrations. The logarithmic renormalization of the velocity was found by Abrikosov and Beneslavsky [31] for the three-dimensional case and in Refs. [33, 32] for two-dimensional graphene. Notice, that no phase transition was revealed even at lowest carrier concentration. We have also to conclude that the Coulomb interactions do not create any gap in the graphen spectrum.
We recall the peculiarity of graphene conductivity in the absence of the magnetic field [34, 35]. For the optical frequency range, when the spacial dispersion of conductivity is not significant, the intraband electron transitions make a contribution
rintra(w) =
2ie2T / u \
——-T- in 2 cosh —- ,
7Thiuj + ir-1) V 2 TJ'
(2)
which has the Drude-Boltzmann form at the large chemical potential i ^ T.
At the zero temperature, the interband electron contribution can be presented in the simple form
r
H=4H
i , (u + 2u)
6(uj - 2u)--n P!0
K W 2TT (w - 2¿t 2
where the 0-function expresses the threshold behavior of interband electron transitions at w = 2^. The temperature smooths out all the singularities in this formula. In high frequency region w ^ (T, ¡i), the interband transitions make the leading contribution into conductivity
r(u) =
4ft'
having the universal character independent of any material parameters. This frequency region is limited above by the band width of around 3 eV.
Making use the universal conductivity, one can calculate the light transmission through graphene [36, 37] in the approximation linear in conductivity
4n e2
T = 1--Re cr(cu) cos 6 = 1 — 7r— cos 8,
c nc
where 0 is the incidence angle of light. In excellent agreement with the theory, for the wide optical range, several experimental groups [38-40] observe the light transmission through graphene as well as bilayer graphene where
2
a
2
e
2
v
the difference from unity is twice as larger. It is exceptionally intriguing that the light transmission involves the fine structure constant a = e2/he of quantum electrodynamics having really no relations to the graphene physics.
For the frequency range, where the intraband term plays the main role, the plasmon excitations are possible [34, 41] with the dispersion
\J~Kk,
2e2T
h2
ln
2 cosh—) 2 TJ
and relatively small damping, determined by the electron relaxation t-1. The plasmon has the same dispersion, h1'2, as the normal 2d plasmon. However, it shows the temperature dependence at low carrier concentrations, n < 2T.
Electron dispersion in bilayer graphene. Bilayer graphene has attracted much interest partly due to the opening of a tunable gap in its electronic spectrum with an external electrostatic field. Such a phenomenon was predicted in Refs. [42, 43] and was observed in optical studies controlled by applying a gate voltage [44-51].
The Hamiltonian of the SWMC theory can be written [24, 25] near the K points in the Brillouin zone in the form
H (p)
U
vp— Yi
vP+ U
Y4P+/YQ
Yi
Y4vp—/yq U
Y4vp—/yq ^ Y3vp+/Y0 vp—
\ Y4vp+ /yq Y3vp—/YQ vp+
U
(3)
/
where p± = ^ipx — py. The nearest-neighbor hopping integral 70 ~ 3 eV corresponds with the velocity parameter v = 1.5ao7o = 106 m/s and the in-layer inter-atomic distance a0 = 1.415 A. The parameters 73,4 ~ 0.1 eV describe the interlayer interaction at the distance d0 = 3.35 A between layers.
Hamiltonian (3) give four levels labeled by the number s = 1,2,3,4 from the bottom. For U = 0, the twofold degeneration e2 = e3 exists at px = py = 0, as a consequence o
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